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Steps to find the irrational numbers between two numbers:
Step 1: Express the provided number in decimal form.
Step 2: Keep in mind that we need to construct irrational numbers between the provided numbers.
Step 3: Within the possible range construct irrational numbers such that they are non-recurring and non-terminating.
Example:
Consider two numbers \(1/6\) and \(2/3\).
The decimal expressions of provided numbers are \(1/6 = 0.1666...\) and \(2/3 = 0.6666...\)
Now we have to construct irrational between \(0.1666...\) and \(0.6666...\)
Remember that irrationals are non-recurring and non-terminating.
We ourself can construct numbers with the given conditions.
Let the numbers are \(0.17070070007....\), \( 0.23003000300003...\), \(0.35010011011101111... \)
In this way, we can construct an infinite number of irrationals between any two numbers.
The easiest way to express an irrational number between two rational numbers:
Step 1: Express the smaller rational number as a terminating decimal.
Step 2: Put \(01\) after its last decimal.
Step 3: Then put \(001\) then put \(0001\) then put \(00001\) … … … so on and on.
Step 4: Hence the resultant number is an irrational number between the two given rational numbers.
Example:
Consider the same two numbers \(1/6\) and \(2/3\).
The decimal expressions of provided numbers are \(1/6 = 0.1666...\) and \(2/3 = 0.6666...\)
Now we have to construct irrational between \(0.1666...\) and \(0.6666...\)
Let us round off the smallest rational number to three decimal places.
Thus, the smallest rational number is \(0.167\).
Now add the number \(01\), it becomes \(0.16701\).
Then add \(001\), \(0.16701001\).
Now add \(0001\), \(0.167010010001\)
Then add \(00001\), \(0.16701001000100001\).
Proceeding in the same way, the number becomes as follows:
\(0.16701001000100001000001...\)
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